https://github.com/cran/bayestestR
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<body>
<h1 id="summary-of-mediation-analysis-using-bayesian-regression-models">Summary of Mediation Analysis using Bayesian Regression Models</h1>
<p>This vignettes demonstrates the <code>mediation()</code>-function. Before we start, we fit some models, including a mediation-object from the <em>mediation</em>-package, which we use for comparison with <em>brms</em> and <em>rstanarm</em>.</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1"></a><span class="kw">library</span>(bayestestR)</span>
<span id="cb1-2"><a href="#cb1-2"></a><span class="kw">library</span>(mediation)</span>
<span id="cb1-3"><a href="#cb1-3"></a><span class="kw">library</span>(brms)</span>
<span id="cb1-4"><a href="#cb1-4"></a><span class="kw">library</span>(rstanarm)</span>
<span id="cb1-5"><a href="#cb1-5"></a></span>
<span id="cb1-6"><a href="#cb1-6"></a><span class="co"># load sample data</span></span>
<span id="cb1-7"><a href="#cb1-7"></a><span class="kw">data</span>(jobs)</span>
<span id="cb1-8"><a href="#cb1-8"></a></span>
<span id="cb1-9"><a href="#cb1-9"></a><span class="kw">set.seed</span>(<span class="dv">123</span>)</span>
<span id="cb1-10"><a href="#cb1-10"></a><span class="co"># linear models, for mediation analysis</span></span>
<span id="cb1-11"><a href="#cb1-11"></a>b1 <-<span class="st"> </span><span class="kw">lm</span>(job_seek <span class="op">~</span><span class="st"> </span>treat <span class="op">+</span><span class="st"> </span>econ_hard <span class="op">+</span><span class="st"> </span>sex <span class="op">+</span><span class="st"> </span>age, <span class="dt">data =</span> jobs)</span>
<span id="cb1-12"><a href="#cb1-12"></a>b2 <-<span class="st"> </span><span class="kw">lm</span>(depress2 <span class="op">~</span><span class="st"> </span>treat <span class="op">+</span><span class="st"> </span>job_seek <span class="op">+</span><span class="st"> </span>econ_hard <span class="op">+</span><span class="st"> </span>sex <span class="op">+</span><span class="st"> </span>age, <span class="dt">data =</span> jobs)</span>
<span id="cb1-13"><a href="#cb1-13"></a></span>
<span id="cb1-14"><a href="#cb1-14"></a><span class="co"># mediation analysis, for comparison with brms</span></span>
<span id="cb1-15"><a href="#cb1-15"></a>m1 <-<span class="st"> </span><span class="kw">mediate</span>(b1, b2, <span class="dt">sims =</span> <span class="dv">1000</span>, <span class="dt">treat =</span> <span class="st">"treat"</span>, <span class="dt">mediator =</span> <span class="st">"job_seek"</span>)</span></code></pre></div>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1"></a><span class="co"># Fit Bayesian mediation model in brms</span></span>
<span id="cb2-2"><a href="#cb2-2"></a>f1 <-<span class="st"> </span><span class="kw">bf</span>(job_seek <span class="op">~</span><span class="st"> </span>treat <span class="op">+</span><span class="st"> </span>econ_hard <span class="op">+</span><span class="st"> </span>sex <span class="op">+</span><span class="st"> </span>age)</span>
<span id="cb2-3"><a href="#cb2-3"></a>f2 <-<span class="st"> </span><span class="kw">bf</span>(depress2 <span class="op">~</span><span class="st"> </span>treat <span class="op">+</span><span class="st"> </span>job_seek <span class="op">+</span><span class="st"> </span>econ_hard <span class="op">+</span><span class="st"> </span>sex <span class="op">+</span><span class="st"> </span>age)</span>
<span id="cb2-4"><a href="#cb2-4"></a>m2 <-<span class="st"> </span><span class="kw">brm</span>(f1 <span class="op">+</span><span class="st"> </span>f2 <span class="op">+</span><span class="st"> </span><span class="kw">set_rescor</span>(<span class="ot">FALSE</span>), <span class="dt">data =</span> jobs, <span class="dt">cores =</span> <span class="dv">4</span>)</span></code></pre></div>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1"></a><span class="co"># Fit Bayesian mediation model in rstanarm</span></span>
<span id="cb3-2"><a href="#cb3-2"></a>m3 <-<span class="st"> </span><span class="kw">stan_mvmer</span>(</span>
<span id="cb3-3"><a href="#cb3-3"></a> <span class="kw">list</span>(job_seek <span class="op">~</span><span class="st"> </span>treat <span class="op">+</span><span class="st"> </span>econ_hard <span class="op">+</span><span class="st"> </span>sex <span class="op">+</span><span class="st"> </span>age <span class="op">+</span><span class="st"> </span>(<span class="dv">1</span> <span class="op">|</span><span class="st"> </span>occp),</span>
<span id="cb3-4"><a href="#cb3-4"></a> depress2 <span class="op">~</span><span class="st"> </span>treat <span class="op">+</span><span class="st"> </span>job_seek <span class="op">+</span><span class="st"> </span>econ_hard <span class="op">+</span><span class="st"> </span>sex <span class="op">+</span><span class="st"> </span>age <span class="op">+</span><span class="st"> </span>(<span class="dv">1</span> <span class="op">|</span><span class="st"> </span>occp)),</span>
<span id="cb3-5"><a href="#cb3-5"></a> <span class="dt">data =</span> jobs,</span>
<span id="cb3-6"><a href="#cb3-6"></a> <span class="dt">cores =</span> <span class="dv">4</span>,</span>
<span id="cb3-7"><a href="#cb3-7"></a> <span class="dt">refresh =</span> <span class="dv">0</span></span>
<span id="cb3-8"><a href="#cb3-8"></a>)</span></code></pre></div>
<p><code>mediation()</code> is a summary function, especially for mediation analysis, i.e. for multivariate response models with casual mediation effects.</p>
<p>In the models <code>m2</code> and <code>m3</code>, <code>treat</code> is the treatment effect and <code>job_seek</code> is the mediator effect. For the <em>brms</em> model (<code>m2</code>), <code>f1</code> describes the mediator model and <code>f2</code> describes the outcome model. This is similar for the <em>rstanarm</em> model.</p>
<p><code>mediation()</code> returns a data frame with information on the <em>direct effect</em> (median value of posterior samples from treatment of the outcome model), <em>mediator effect</em> (median value of posterior samples from mediator of the outcome model), <em>indirect effect</em> (median value of the multiplication of the posterior samples from mediator of the outcome model and the posterior samples from treatment of the mediation model) and the <em>total effect</em> (median value of sums of posterior samples used for the direct and indirect effect). The <em>proportion mediated</em> is the indirect effect divided by the total effect.</p>
<p>The simplest call just needs the model-object.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1"></a><span class="co"># for brms</span></span>
<span id="cb4-2"><a href="#cb4-2"></a><span class="kw">mediation</span>(m2)</span>
<span id="cb4-3"><a href="#cb4-3"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb4-4"><a href="#cb4-4"></a><span class="co">#> </span></span>
<span id="cb4-5"><a href="#cb4-5"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb4-6"><a href="#cb4-6"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb4-7"><a href="#cb4-7"></a><span class="co">#> Response : depress2</span></span>
<span id="cb4-8"><a href="#cb4-8"></a><span class="co">#> </span></span>
<span id="cb4-9"><a href="#cb4-9"></a><span class="co">#> Effect | Estimate | 89% ETI</span></span>
<span id="cb4-10"><a href="#cb4-10"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb4-11"><a href="#cb4-11"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.110, 0.031]</span></span>
<span id="cb4-12"><a href="#cb4-12"></a><span class="co">#> Indirect Effect (ACME) | -0.015 | [-0.036, 0.004]</span></span>
<span id="cb4-13"><a href="#cb4-13"></a><span class="co">#> Mediator Effect | -0.240 | [-0.285, -0.195]</span></span>
<span id="cb4-14"><a href="#cb4-14"></a><span class="co">#> Total Effect | -0.055 | [-0.129, 0.018]</span></span>
<span id="cb4-15"><a href="#cb4-15"></a><span class="co">#> </span></span>
<span id="cb4-16"><a href="#cb4-16"></a><span class="co">#> Proportion mediated: 28.14% [-71.11%, 127.40%]</span></span>
<span id="cb4-17"><a href="#cb4-17"></a></span>
<span id="cb4-18"><a href="#cb4-18"></a><span class="co"># for rstanarm</span></span>
<span id="cb4-19"><a href="#cb4-19"></a><span class="kw">mediation</span>(m3)</span>
<span id="cb4-20"><a href="#cb4-20"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb4-21"><a href="#cb4-21"></a><span class="co">#> </span></span>
<span id="cb4-22"><a href="#cb4-22"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb4-23"><a href="#cb4-23"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb4-24"><a href="#cb4-24"></a><span class="co">#> Response : depress2</span></span>
<span id="cb4-25"><a href="#cb4-25"></a><span class="co">#> </span></span>
<span id="cb4-26"><a href="#cb4-26"></a><span class="co">#> Effect | Estimate | 89% ETI</span></span>
<span id="cb4-27"><a href="#cb4-27"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb4-28"><a href="#cb4-28"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.111, 0.031]</span></span>
<span id="cb4-29"><a href="#cb4-29"></a><span class="co">#> Indirect Effect (ACME) | -0.018 | [-0.037, 0.002]</span></span>
<span id="cb4-30"><a href="#cb4-30"></a><span class="co">#> Mediator Effect | -0.241 | [-0.286, -0.197]</span></span>
<span id="cb4-31"><a href="#cb4-31"></a><span class="co">#> Total Effect | -0.057 | [-0.130, 0.017]</span></span>
<span id="cb4-32"><a href="#cb4-32"></a><span class="co">#> </span></span>
<span id="cb4-33"><a href="#cb4-33"></a><span class="co">#> Proportion mediated: 30.59% [-75.65%, 136.82%]</span></span></code></pre></div>
<p>Typically, <code>mediation()</code> finds the treatment and mediator variables automatically. If this does not work, use the <code>treatment</code> and <code>mediator</code> arguments to specify the related variable names. For all values, the 89% credible intervals are calculated by default. Use <code>ci</code> to calculate a different interval.</p>
<p>Here is a comparison with the <em>mediation</em> package. Note that the <code>summary()</code>-output of the <em>mediation</em> package shows the indirect effect first, followed by the direct effect.</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1"></a><span class="kw">summary</span>(m1)</span>
<span id="cb5-2"><a href="#cb5-2"></a><span class="co">#> </span></span>
<span id="cb5-3"><a href="#cb5-3"></a><span class="co">#> Causal Mediation Analysis </span></span>
<span id="cb5-4"><a href="#cb5-4"></a><span class="co">#> </span></span>
<span id="cb5-5"><a href="#cb5-5"></a><span class="co">#> Quasi-Bayesian Confidence Intervals</span></span>
<span id="cb5-6"><a href="#cb5-6"></a><span class="co">#> </span></span>
<span id="cb5-7"><a href="#cb5-7"></a><span class="co">#> Estimate 95% CI Lower 95% CI Upper p-value</span></span>
<span id="cb5-8"><a href="#cb5-8"></a><span class="co">#> ACME -0.0157 -0.0387 0.01 0.19</span></span>
<span id="cb5-9"><a href="#cb5-9"></a><span class="co">#> ADE -0.0438 -0.1315 0.04 0.35</span></span>
<span id="cb5-10"><a href="#cb5-10"></a><span class="co">#> Total Effect -0.0595 -0.1530 0.02 0.21</span></span>
<span id="cb5-11"><a href="#cb5-11"></a><span class="co">#> Prop. Mediated 0.2137 -2.0277 2.70 0.32</span></span>
<span id="cb5-12"><a href="#cb5-12"></a><span class="co">#> </span></span>
<span id="cb5-13"><a href="#cb5-13"></a><span class="co">#> Sample Size Used: 899 </span></span>
<span id="cb5-14"><a href="#cb5-14"></a><span class="co">#> </span></span>
<span id="cb5-15"><a href="#cb5-15"></a><span class="co">#> </span></span>
<span id="cb5-16"><a href="#cb5-16"></a><span class="co">#> Simulations: 1000</span></span>
<span id="cb5-17"><a href="#cb5-17"></a></span>
<span id="cb5-18"><a href="#cb5-18"></a><span class="kw">mediation</span>(m2, <span class="dt">ci =</span> <span class="fl">.95</span>)</span>
<span id="cb5-19"><a href="#cb5-19"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb5-20"><a href="#cb5-20"></a><span class="co">#> </span></span>
<span id="cb5-21"><a href="#cb5-21"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb5-22"><a href="#cb5-22"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb5-23"><a href="#cb5-23"></a><span class="co">#> Response : depress2</span></span>
<span id="cb5-24"><a href="#cb5-24"></a><span class="co">#> </span></span>
<span id="cb5-25"><a href="#cb5-25"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb5-26"><a href="#cb5-26"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb5-27"><a href="#cb5-27"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.124, 0.046]</span></span>
<span id="cb5-28"><a href="#cb5-28"></a><span class="co">#> Indirect Effect (ACME) | -0.015 | [-0.041, 0.008]</span></span>
<span id="cb5-29"><a href="#cb5-29"></a><span class="co">#> Mediator Effect | -0.240 | [-0.294, -0.185]</span></span>
<span id="cb5-30"><a href="#cb5-30"></a><span class="co">#> Total Effect | -0.055 | [-0.145, 0.034]</span></span>
<span id="cb5-31"><a href="#cb5-31"></a><span class="co">#> </span></span>
<span id="cb5-32"><a href="#cb5-32"></a><span class="co">#> Proportion mediated: 28.14% [-181.46%, 237.75%]</span></span>
<span id="cb5-33"><a href="#cb5-33"></a></span>
<span id="cb5-34"><a href="#cb5-34"></a><span class="kw">mediation</span>(m3, <span class="dt">ci =</span> <span class="fl">.95</span>)</span>
<span id="cb5-35"><a href="#cb5-35"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb5-36"><a href="#cb5-36"></a><span class="co">#> </span></span>
<span id="cb5-37"><a href="#cb5-37"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb5-38"><a href="#cb5-38"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb5-39"><a href="#cb5-39"></a><span class="co">#> Response : depress2</span></span>
<span id="cb5-40"><a href="#cb5-40"></a><span class="co">#> </span></span>
<span id="cb5-41"><a href="#cb5-41"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb5-42"><a href="#cb5-42"></a><span class="co">#> ----------------------------------------------------</span></span>
<span id="cb5-43"><a href="#cb5-43"></a><span class="co">#> Direct Effect (ADE) | -0.040 | [-0.129, 0.048]</span></span>
<span id="cb5-44"><a href="#cb5-44"></a><span class="co">#> Indirect Effect (ACME) | -0.018 | [-0.042, 0.006]</span></span>
<span id="cb5-45"><a href="#cb5-45"></a><span class="co">#> Mediator Effect | -0.241 | [-0.296, -0.187]</span></span>
<span id="cb5-46"><a href="#cb5-46"></a><span class="co">#> Total Effect | -0.057 | [-0.151, 0.033]</span></span>
<span id="cb5-47"><a href="#cb5-47"></a><span class="co">#> </span></span>
<span id="cb5-48"><a href="#cb5-48"></a><span class="co">#> Proportion mediated: 30.59% [-221.09%, 282.26%]</span></span></code></pre></div>
<p>If you want to calculate mean instead of median values from the posterior samples, use the <code>centrality</code>-argument. Furthermore, there is a <code>print()</code>-method, which allows to print more digits.</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1"></a>m <-<span class="st"> </span><span class="kw">mediation</span>(m2, <span class="dt">centrality =</span> <span class="st">"mean"</span>, <span class="dt">ci =</span> <span class="fl">.95</span>)</span>
<span id="cb6-2"><a href="#cb6-2"></a><span class="kw">print</span>(m, <span class="dt">digits =</span> <span class="dv">4</span>)</span>
<span id="cb6-3"><a href="#cb6-3"></a><span class="co">#> # Causal Mediation Analysis for Stan Model</span></span>
<span id="cb6-4"><a href="#cb6-4"></a><span class="co">#> </span></span>
<span id="cb6-5"><a href="#cb6-5"></a><span class="co">#> Treatment: treat</span></span>
<span id="cb6-6"><a href="#cb6-6"></a><span class="co">#> Mediator : job_seek</span></span>
<span id="cb6-7"><a href="#cb6-7"></a><span class="co">#> Response : depress2</span></span>
<span id="cb6-8"><a href="#cb6-8"></a><span class="co">#> </span></span>
<span id="cb6-9"><a href="#cb6-9"></a><span class="co">#> Effect | Estimate | 95% ETI</span></span>
<span id="cb6-10"><a href="#cb6-10"></a><span class="co">#> ------------------------------------------------------</span></span>
<span id="cb6-11"><a href="#cb6-11"></a><span class="co">#> Direct Effect (ADE) | -0.0395 | [-0.1237, 0.0456]</span></span>
<span id="cb6-12"><a href="#cb6-12"></a><span class="co">#> Indirect Effect (ACME) | -0.0158 | [-0.0405, 0.0083]</span></span>
<span id="cb6-13"><a href="#cb6-13"></a><span class="co">#> Mediator Effect | -0.2401 | [-0.2944, -0.1846]</span></span>
<span id="cb6-14"><a href="#cb6-14"></a><span class="co">#> Total Effect | -0.0553 | [-0.1454, 0.0341]</span></span>
<span id="cb6-15"><a href="#cb6-15"></a><span class="co">#> </span></span>
<span id="cb6-16"><a href="#cb6-16"></a><span class="co">#> Proportion mediated: 28.60% [-181.01%, 238.20%]</span></span></code></pre></div>
<p>As you can see, the results are similar to what the <em>mediation</em> package produces for non-Bayesian models.</p>
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